Moreover, \(f^{(\lambda ,\phi )}\)  is nonnegative, lower-semicontinuous and for all \((t,x) \in [0,T] \times \mathbb {R}^d\) , \((q,N) \rightarrow f^{(\lambda ,\phi )}(t,x,q,N) \)  is convex so we can proceed as in [1]} Theorem 2.34 and Example 2.36 to prove that \(\omega ,W\)  are absolutely continuous with respect to \(m\)  and
\( \mathcal {L}_2( ( \lambda ,\phi ),( m,\omega ,W) ) \le \liminf _{k \rightarrow + \infty } \mathcal {L}_2( ( \lambda ,\phi ), (m,\omega ,W) ). \) 
